Spaces with metric that takes values in partially ordered vector spaces were introduced by Kurepa [1]} in 1934. Later Kantorovich developed a theory of normed spaces with norms that take values in complete vector latices, see [2]}. These objects appeared to be fruitful in the study of functional equations using iterative methods, and in related questions of Analysis, see [3]}.
Results and references on the fixed point theorems for the metric spaces with metric that takes values in a partially ordered vector spaces (\(K\) -metric spaces and cone-metric spaces) can be found in surveys [4]}, [5]}. In particular, in these articles the question whether results on cone metric or \(K\) -metric spaces can be reduced to results on ordinary metric spaces, is discussed.
Parametric metric spaces [6]}, probabilistic metric spaces [7]}, [8]}, fuzzy metric spaces [7]}, [10]} and intuitionistic metric spaces (see for references [6]}, [12]}), gauge spaces (see [13]}) also were investigated in Fixed Point Theory.
